Plotting Complex Functions.

I intend to do a taylor series expansion at t=0. I would like to plot the coefficients of each Taylor term in the complex plane, using the surf function.

I tried Taylor(F) , Series (F) somewhere there is an error. The function is product of two gaussians, with two parameters mentioned above. This Gaussian contains other constants, Some are symbolic, while some have pre determined values.

Can some one help me out.

Thanks.

[Torsten]

You mean

alpha = alpha_r + 1i * alpha_i

with alpha_r, alpha_i being real numbers (not both of them 0) ?

Note that F is undefined at t = 0, the point at which you want to compute the Taylor expansion.

[John D'Errico]

You have multiple constants in there with no value given. You say this yourself. You cannot plot something that is unkown.

At best, you can assign values to ALL of the constants, then plot to your hearts content.

[David Goodmanson]

Since exp(-inf) = 0, I believe that F is well defined with F= 0 at t=0. A person might want to define it as a limit for increasingly large values of t, but that limit is 0. Unlike, say, sin(1/t) as t -->0.

Answer 1

[David Goodmanson]

Hello Sudharsana,

You can take the product of both functions to obtain either

f(t) = exp((-ar-ai)/(twC))

or

f(t) = exp((-ar-i*ai)/(twC))

depending on what is meant (per Torsten's comment). Either way you can reduce this to

f(t) = exp(-b/t)

for some (possibly complex) constant b. This function has an essential singularity at t = 0. The function and all its derivatives are zero at t = 0. Since the coefficients of a Taylor series about t = 0 are determined by the derivatives of f(t) at that point, this means that f(t) cannot be expanded in a Taylor series about t = 0.

Which doesn't mean that it can't be approximated. For example, using b = 1 for simplicity

t = 0:.001:4;
f = exp(-1./t);
c = polyfit(t,f,6)
f_fit = polyval(c,t);

I'm not advocating for polyfit which is anyway not a great idea for polynomials of large degree, and I am sure there are better fits but this is just one example.